Thin Film Materials-L. B. Freund,S. Suresh

1.8.1 Compressive stress prior to island coalescence 711.8.5 Correlations between ¯nal stress and grain structure 78 2.2.1 Substrate curvature for arbitrary ¯lm thickness 1032.2.2 Exampl

Stress, Defect Formation and Surface Evolution

L B FreundDivision of Engineering Brown University

S SureshDepartment of Materials Science and Engineering Massachusetts Institute of Technology

(January 12, 2003)

page xiii

1.3.3 Energy density of a free surface or an interface 22

1.5.2 The damascene process for copper interconnects 54

iv

1.8.1 Compressive stress prior to island coalescence 71

1.8.5 Correlations between ¯nal stress and grain structure 78

2.2.1 Substrate curvature for arbitrary ¯lm thickness 1032.2.2 Example: Maximum thermal stress in a bilayer 110

2.4.1 Nonuniform mismatch strain and elastic properties 126

2.4.3 Example: Stress in compositionally graded ¯lms 131

2.4.5 Example: Overall thermoelastic response of a multilayer 134

2.5.2 Axially symmetric deformation in the nonlinear range 141

2.6.1 Bifurcation analysis with uniform curvature 1462.6.2 Visualization of states of uniform curvature 1542.6.3 Bifurcation for general curvature variation 158

2.6.5 Example: A curvature map for a Cu/Si system 163

3.5.2 Example: Cubic thin ¯lm with a (111) orientation 184

3.6.2 Example: Stress implied by measured d¡spacing 1883.6.3 Stress-free d¡spacing from asymmetric di®raction 1893.6.4 Example: Determination of reference lattice spacing 194

3.7.1 Anisotropic thin ¯lm on an isotropic substrate 195

3.8.2 Example: Substrate curvature due to an electric ¯eld 203

3.9.1 Plane strain curvature change due to ¯lm cracks 206

3.10.2 Volume averaged stress in terms of curvature 2243.10.3 Volume averaged stress in a damascene structure 227

4 Delamination and fracture 239

4.1.2 Example: An equation governing interfacial shear stress 245

4.2.1 Energy release rate and the Gri±th criterion 2544.2.2 Example: Interface toughness of a laminated composite 259

4.3.1 Characterization of interface separation behavior 2694.3.2 E®ects of processing and interface chemistry 2724.3.3 E®ect of local phase angle on fracture energy 276

4.4.2 Example: Delamination due to thermal strain 287

4.4.4 Phase angle of the stress concentration ¯eld 293

5.1.2 Driving force for growth of delamination 3495.1.3 Phase angle of local stress state at interface 350

5.2.2 Example: Temperature change for buckling of a debond 363

5.4.3 E®ects of imperfections on buckling delamination 3775.4.4 Example: Buckling instability of carbon ¯lms 380

5.5.3 Example: Buckling wavelength for a glass substrate 387

6.1.2 Elastic ¯eld of a dislocation near a free surface 426

6.2 Critical thickness of a strained epitaxial ¯lm 432

6.2.2 Dependence of critical thickness on mismatch strain 4366.2.3 Example: Critical thickness of a SiGe ¯lm on Si(001) 4386.2.4 Experimental results for critical thickness 4396.2.5 Example: In°uence of crystallographic orientation on hcr 441

6.3.1 Condition for advance of a threading dislocation 4446.3.2 Limitations of the critical thickness condition 4486.3.3 Threading dislocation under nonequilibrium conditions 451

6.4.1 Uniform strained layer capped by an unstrained layer 456

6.5.1 Critical thickness condition for the model system 4646.5.2 The in°uence of ¯lm{substrate modulus di®erence 4656.5.3 Example: Modulus di®erence and dislocation formation 469

6.6.2 E®ect of a free surface on quantum wire stability 477

6.7.2 Example: Critical thickness for a compliant substrate 4866.7.3 Mis¯t strain relaxation due to a viscous underlayer 487

6.8.1 Spontaneous formation of a surface dislocation loop 4956.8.2 Dislocation nucleation in a perfect crystal 4976.8.3 E®ect of a stress concentrator on nucleation 501

7.1.2 Spacing for simultaneous formation of dislocations 5097.1.3 Spacing based on insertion of the last dislocation 511

7.3 Strain relaxation due to dislocation formation 523

7.3.2 Example: Dislocation control in semiconductor ¯lms 527

7.5.2 Example: Temperature cycling with isotropic hardening 546

7.6.1 Thermally activated dislocation glide past obstacles 555

7.7.1 Experimental observation of grain structure evolution 5617.7.2 Experimental observation of threading dislocations 5637.7.3 Strain relaxation mechanisms during temperature cycling 566

7.9.3 Example: Thin ¯lm undergoing plane strain extension 589

8.2.2 Mass transport along a bimaterial interface 607

8.4 Periodic perturbation of a °at surface 624

8.4.2 Example: Stability of a strained epitaxial ¯lm 6298.4.3 In°uence of substrate sti®ness on surface stability 630

8.4.5 Example: Validity of the small slope approximation 638

8.5.3 Example: Doubly periodic surface perturbation 642

8.6.1 Force{de°ection relationship for spherical surfaces 6448.6.2 Example: Stress generated when islands impinge 650

8.7.2 Growth patterning due to mis¯t dislocations 654

8.8.4 Example: Stepped surface near (001) for strained Si 669

8.9.4 Nucleation barrier for islands on stable surfaces 6848.9.5 Shape transition for preferred sidewall orientations 686

9.3.1 A variational principle for surface °ux 7169.3.2 Application to second order surface perturbation 720

9.5.1 Stress relaxation by grain boundary di®usion 7329.5.2 Di®usion along shear bands during deformation 738

9.6.3 Example: Elastic stabilization of a composition 749

9.7.3 E®ects of microstructure on electromigration damage 767

Within a period of a few decades, the ¯eld of materials science and neering has emerged as a focal point for developments in virtually all areas

engi-of engineering and applied science The study engi-of thin film materials hasbeen one of the unifying themes in the development of the ¯eld during thisperiod As understood here, the area encompasses ¯lms bonded to rela-tively thick substrates, multilayer materials, patterned ¯lms on substratesand free-standing ¯lms Signi¯cant advances in methods for synthesizingand processing these materials for ever more speci¯c purposes, as well as ininstrumentation for characterizing materials at ever diminishing size scales,have been key to modern engineering progress

At the dawn of the 21st century, the United States National Academy

of Engineering reported the outcome of a project intended to identify thetwenty most signi¯cant engineering achievements of the preceding century

It is evident from the list compiled that achievements of the second half

of the twentieth century { electronics, computers, health technologies, laserand ¯ber optics, for example { were all based on the creative and e±cientexploitation of materials; thin ¯lm materials represent a major component

of this advance in materials technology In fact, the impact of advances

in the specialized uses of materials was so pervasive in the achievementsbeing recognized by the Academy that the development of high-performancematerials itself was included as one of the most signi¯cant achievements.The goal of this book is to summarize developments in the area of thin

¯lm materials that have occurred over the past few decades, with emphasis

on the generation of internal stress and its consequences Internal stress caninduce a variety of undesirable consequences including excessive deforma-tion, fracture, delamination, permanent deformation and microstructuralalterations In spite of these possibilities, thin ¯lms have been insertedinto engineering systems in order to accomplish a wide range of practical

xv

service functions Among these are microelectronic devices and packages;micro-electro-mechanical systems or MEMS; and surface coatings intended

to impart a thermal, mechanical, tribological, environmental, optical, trical, magnetic or biological function To a large extent, the success of thisendeavor has been enabled by research leading to reliable means for esti-mating stress in small material systems and by establishing frameworks inwhich to assess the integrity or functionality of the systems The prospectfor material failure due to stress continues to be a technology-limiting bar-rier, even in situations in which load-carrying capacity of the material is notamong its primary functional characteristics In some circumstances, stresshas desirable consequences, as in bandgap engineering for electronic appli-cations and in the self-assembly of small structures driven by stored elasticenergy It is our hope that the information included in this book will beuseful as an indicator of achievements in the ¯eld and as a guide for furtheradvances in a number of new and emerging directions

elec-The ¯rst chapter is devoted largely to a discussion of the origins ofresidual stress in thin ¯lm materials and to identi¯cation of relationshipsbetween processing methods and generation of stress The consequences ofstress are discussed in subsequent chapters, with the presentation generallyorganized according to the size scale of the dominant physical phenomenainvolved Overall deformation of ¯lm-substrate systems or multilayer struc-tures are considered in Chapters 2 and 3 This is followed by examination

of the general failure modes of fracture, delamination and buckling of ¯lms

in Chapters 4 and 5 The focus then shifts to a smaller scale to discussconditions for dislocation formation in Chapter 6 and inelastic deformation

of ¯lms in Chapter 7 Finally, the issues of stability of material surfacesand evolution of surface morphology or alloy composition are considered inChapters 8 and 9 The consequences of stress in thin ¯lms is linked to thestructure of the ¯lm materials wherever possible

It is recognized that each of the principal topics covered in the bookcould itself be developed into a substantial monograph, but the goal here

is not the exhaustive treatment of a topic of limited scope The area isinherently interdisciplinary, and the intention is a provide a comprehensivecoverage of issues relevant to stress and its consequences in thin ¯lm mate-rials Adoption of this approach meant that many choices had to be madealong the way about depth of coverage of speci¯c topics and balance amongdi®erent topics; we hope that the readers will judge the choices made to bereasonable The main purpose of the book is the coherent presentation ofthe sound scienti¯c basis for describing the origins of stress in ¯lms and foranticipating the consequences of stress in defect formation, surface evolution

and allied e®ects Many references to original work are included as a guide

to the archival literature in the area In addition, the fundamental conceptsdeveloped are made more concrete through implementation in sample calcu-lations and through discussion of case studies of practical signi¯cance Thedescription of experimental methods, results and observations is included

as an integral part of developing the conceptual structure of the topics amined Each chapter concludes with a set of exercises that further extendthe material discussed, and which can challenge newcomers to the area atapplying concepts It is our hope that, with this structure, the book willserve as a research reference for those pursuing the area at its frontiers, as auseful compilation of readily applicable results for practicing engineers, and

ex-as a textbook for graduate students or advanced undergraduate studentswishing to develop background in this area

The idea for the book grew out of a course on thin ¯lms that hasbeen o®ered for students in solid mechanics and materials science at BrownUniversity since 1992, as a natural outgrowth of emerging research activity

in the area We are grateful to the many students, postdoctoral researchassociates and colleagues who attended these lectures and whose enthusiasmgave this project its initial impetus

We are also grateful to many colleagues who have contributed in ious ways to the preparation of this book We particularly thank JohnHutchinson who used a draft of parts of the book for a course for graduatestudents at Harvard and MIT, and who provided valuable feedback on thismaterial Both John Hutchinson and Bill Nix kindly shared with us theirown course materials on thin ¯lms Several colleagues read drafts of vari-ous sections of the book and o®ered helpful recommendations; they includeIlan Blech, Eric Chason, Ares Rosakis, Vivek Shenoy and Carl Thompson.Several graduate students who took courses based in part on draft chapters,particularly Yoonjoon Choi and Nuwong Chollacoop, suggested a number ofclari¯cations and improvements in the presentation Finally, we are grateful

var-to the many colleagues who provided ¯gures and micrographs from theirown work; in these cases, acknowledgments are noted along with the in-cluded material Tim Fishlock at Cambridge University Press o®ered usconsiderable °exibility in the formulation of the scope of this book and inthe preparation of the document

LBF is grateful to the Materials Research Science and EngineeringCenter, funded by the National Science Foundation at Brown University,for long-term support of research in the general area of thin ¯lms and forthe collaborations fostered through the Center He is also thankful to theDivision of Engineering and Applied Sciences at Caltech for hosting a sab-

batical leave; the kind hospitality and congenial environment a®orded anopportunity for pursuing the book writing project at its early critical stage.

SS is grateful to the Defense University Research Initiative in ogy, funded by the O±ce of Naval Research at MIT, and the Programme onAdvanced Materials for Micro and Nano Systems, funded by the Singapore-MIT Alliance, for their ¯nancial support for research in the areas covered

NanoTechnol-by the book

A project of this magnitude would not have been possible withoutthe support and encouragement of the members of our families We areextremely grateful for their enduring patience and understanding duringour long hours of immersion in this project over the past several years

L B Freund and S Suresh

January 2003

Introduction and Overview

Thin solid ¯lms have been used in many types of engineering systems andhave been adapted to ful¯ll a wide variety of functions A few examplesfollow

¡ Great strides in thin ¯lm technology have been made in order toadvance the rapid development of miniature, highly integrated elec-tronic circuits In such devices, con¯nement of electric charge relieslargely on interfaces between materials with di®ering electronic prop-erties Furthermore, the need for thin materials of exceptionally highquality, reproducible characteristics and reliability has driven ¯lmgrowth technology through a rapid succession of signi¯cant achieve-ments More recently, progress in the physics of material structuresthat rely on quantum con¯nement of charge carriers continues torevolutionize the area These systems present new challenges for ma-terials synthesis, characterization and modeling

¡ The use of surface coatings to protect structural materials in hightemperature environments is another thin ¯lm technology of enor-mous commercial signi¯cance In gas turbine engines, for example,thin surface ¯lms of materials chosen for their chemical inertness,stability at elevated temperatures and low thermal conductivity areused to increase engine e±ciency and to extend signi¯cantly the usefullifetimes of the structural materials that they protect Multilayed orcontinuously graded coatings o®er the potential for further progress

in this e®ort

¡ The useful lifetimes of components subjected to friction and wear due

to contact can be extended substantially through the use of surface

1

coatings or surface treatments Among the technologies that rely onthe use of thin ¯lms in this way are internal combustion engines, ar-ti¯cial hip and knee implants, and computer hard disks for magneticdata storage.

¡ Thin ¯lms are integral parts of many micro-electro-mechanical tems designed to serve as sensors or actuators For example, a piezo-electric or piezoresistive thin ¯lm deposited on a silicon membranecan be used to detect electronically a de°ection of the membrane inresponse to a pressure applied on its surface or by an acceleration ofits supports Devices based on thin ¯lm technology are used as mi-crophones in hearing aids, monitors of blood pressure during exercise,electronically positioned thin ¯lm mirrors on °exible supports in op-tical display systems, and probes for detecting the degree of ripeness

sys-of fruits

Numerous other technologies rely on thin ¯lm behavior An diate observation that follows from the foregoing list is that the principalfunction of the thin ¯lm components in these applications is often not struc-tural Consequently, load carrying capacity may not be a principal consid-eration for design or material selection However, fabrication of thin ¯lmcon¯gurations typically results in internal stress in the ¯lm of a magnitudesu±cient to induce mechanical deformation, damage or failure A tendencyfor stress-driven failure of a thin ¯lm structure can be a disabling barrier toincorporation of that ¯lm into a system, even when load carrying capacity

imme-is of secondary importance as a functional characterimme-istic The presence of

an internal stress in a thin ¯lm structure may also in°uence the electrical

or magnetic properties in functional devices

This chapter provides an overview of commonly used deposition andprocessing methods for the synthesis and fabrication of thin ¯lm and suchstructures This is followed by a discussion of the fabrication of small vol-ume structures, where examples of the basic steps involved in lithography,surface micromachining, bulk micromachining, and molding processes areconsidered in the context of the manufacture of microelectronic devices,

as well as small structures encountered in the development of mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS).Attention is devoted to the e®ects of processing on the nucleation and growth

micro-electro-of monocrystalline and polycrystalline thin ¯lms on substrates, the evolution

of ¯lm microstructure in polycrystalline ¯lms, and the generation of nal stresses through processing In the chapters that follow, consequences of

inter-stress in thin ¯lm and multilayer materials is studied, organized according

to the deformation or failure phenomena which can be induced by internalstress

1.1 A classification of thin film configurations

As a guide for the development and application of concepts that are usefulfor describing mechanical behavior of solid thin ¯lm and multilayer ma-terials, it is convenient to classify structures in terms of their geometricalcon¯gurations and the nature of constraint on their deformation imposed bytheir surroundings For this purpose, con¯gurations are classi¯ed in terms

of the relative extent of the solid bodies in three orthogonal directions, withthe orientation of the reference coordinate system being dictated by the con-

¯guration The degree of constraint is determined by the interaction of thethin ¯lm structure with other deformable solids to which it may be attached,

or with which it may otherwise be in contact The former situation requirescompatibility of deformation, while the latter requires some restriction onits motion The categories of con¯guration are termed ¯lm (or layer), line(or wire), and island (or dot); the categories of constraint are termed uncon-

¯ned, partially con¯ned and fully con¯ned These classes are illustrated inFigure 1.1 There is nothing fundamental about such a classi¯cation scheme,but its adoption can facilitate understanding of the ranges of applicability

of the various ideas in the ¯eld

With reference to Figure 1.1, a structure of an extent that is small inone direction compared to its extent in the other two directions is termed

a thin ¯lm; in structural mechanics, such con¯gurations are identi¯ed asplates or shells The quali¯er `small' as used here means that the largestdimensions are at least twenty times greater than the small dimension, andmore commonly are hundreds of times greater than the small dimension Astructure that has small extent in two directions compared to its extent inthe third direction is termed a line or wire; such con¯gurations are usuallyidenti¯ed as rods or bars in structural mechanics Lastly, a structure thathas small extent in all three directions, compared to the dimensions of itssurroundings in this case, is termed an island or a dot

Concerning the degree of constraint on deformation, a small structure

is said to be uncon¯ned if the boundaries associated with its thin sions are free to displace without restriction On the other hand, it is said

dimen-to be fully con¯ned if all boundaries associated with its thin dimensions areconstrained against deformation In virtually all cases, the constraint at aboundary is due to another material which shares that boundary as a com-

mon interface The structure is said to be partially con¯ned if displacement

of its boundaries associated with some, but not all, directions of thinnessare unconstrained

The classi¯cation matrix in Figure 1.1 includes some illustrations ofvarying degrees of con¯nement and thinness As a speci¯c example, consider

a layer of a SiGe alloy 1 ¹m in thickness, which is deposited on a 1 cm by

1 cm area of a Si substrate that is 0:5 mm thick This con¯guration results

in a partially con¯ned thin ¯lm structure A stripe of copper with a squarecross-section that is 0:5 ¹m on a side and a length of 5 mm deposited on

a relatively thick Si substrate is a partially con¯ned line If the surfaces

of the Si substrate and the wire are then completely covered over with a

1 ¹m thick coating of SiO2 to electronically isolate the wire, the structurebecomes a fully con¯ned line An InAs quantum dot being formed by stressdriven surface di®usion is a partially con¯ned island If this con¯guration isthen covered over by a blanket deposit of AlAs, the ¯nal structure is a fullycon¯ned island or quantum dot con¯guration

The idea of the classi¯cation scheme is intended only as a conceptualguide There are many situations in which the structure exhibits behaviorrepresented by more than one entry in the categorization matrix in Fig-

ure 1.1 For example, suppose that a thin ¯lm bonded to a relatively thicksubstrate supports a compressive stress This is a partially con¯ned thin ¯lmcon¯guration If the stress becomes large enough in magnitude, the ¯lm willtend to buckle by debonding from the substrate over some portion of theinterface and then de°ecting away from the substrate over this portion Theregion of the buckle becomes an uncon¯ned ¯lm but the remainder of the

¯lm is partially con¯ned

The categorization of thin ¯lm systems as summarized in Figure 1.1 isbased on relative physical dimensions and makes no reference to any lengthscale re°ecting the underlying structure of the material There is usually aheirarchy of such length scales associated with a material of given chemicalcomposition, and the scales can depend on the processing methods used toform the material structure For example, for a polycrystalline ¯lm, intrinsiclength scales include the size of the atomic unit cell, the spacing of crystallinedefects, and the size of the crystal grains, at the very least Thus, a furthersubcategorization of ¯lm con¯gurations based on a comparison of the smalldimension of the thin structure to the absolute length scale characteristic ofthe constituent material must be considered

When the thickness of the ¯lm is small compared to that of the strate (typically by a factor of 50 or more), it represents a mechanically thinfilm In this case, the ¯lm material either has no intrinsic structural lengthscales, as in the case of an amorphous ¯lm, or the ¯lm thickness is muchlarger than all the characteristic microstructural length scales such as thegrain size, dislocation cell size, precipitate or particle spacing, diameter ofthe dislocation loops, mean free path for dislocation motion, or the magneticdomain wall size Such structures, typically tens or hundreds of microme-ters in thickness, are deposited onto substrates by plasma spray or physicalvapor deposition, or layers bonded to substrates through welding, di®usionbonding, explosion cladding, sintering or self-propagating high temperaturecombustion synthesis This de¯nition, of course, holds only when the sizescale of the microstructure is small compared to the ¯lm thickness Thecontinuum mechanics approach to be presented for the analysis of stress,substrate curvature and fracture in such mechanically thin ¯lms applies to

sub-a brosub-ad rsub-ange of prsub-acticsub-al situsub-ations

When the small dimension of the material structure is comparable tothe characteristic microstructural size scale, the ¯lm is considered to be amicrostructurally thin film Most metallic thin ¯lms used in microelectronicdevices and magnetic storage media are examples of microstructurally thin

¯lms, where the ¯lm thickness is substantially greater than atomic or cular dimensions Although the ¯lm thickness normally includes only a few

mole-microstructural units in these cases, the plane of the ¯lm has dimensions ni¯cantly larger than the characteristic microstructural size scale The me-chanical properties of these ¯lms are much more strongly in°uenced by suchfactors as average grain size, grain shape, grain size distribution, and crystal-lographic texture than in the case of mechanically thin ¯lms Grain to grainvariations in crystallographic orientation as well as crystalline anisotropy ofthermal, electrical, magnetic and mechanical properties also have a morepronounced e®ect on the overall mechanical response of microstructurallythin ¯lms The mechanisms and mechanics of microstructurally thin ¯lmsare considered extensively throughout this book A microstructurally thin

sig-¯lm can be patterned into lines or stripes on the substrate surface, in whichcase the cross-sectional dimensions of each line are comparable to the mi-crostructural unit dimension For single crystal ¯lms epitaxially bonded torelatively thick substrates, the only microstructurally signi¯cant dimension

is the lattice spacing Consequently, such ¯lms can usually be treated asmicrostructurally thin ¯lms even though the ¯lm thickness may be as small

as several times the atomic unit cell dimension Such structures are studied

in Chapter 6

Atomically thin filmsconstitute layers whose thicknesses are ble to one or a few atomic layers An adsorbed monolayer of gas or impurityatoms on a surface is an example of an atomically thin layer Here themechanical response of the thin layer is likely to be more in°uenced by in-teratomic potentials and surface energy than by macroscopic mechanicalproperties or by micromechanisms of deformation

compara-1.2 Film deposition methodsPhysical vapor deposition (PVD) and chemical vapor deposition (CVD) arethe most common methods for transferring material atom by atom fromone or more sources to the growth surface of a ¯lm being deposited onto

a substrate Vapor deposition describes any process in which a solid mersed in a vapor becomes larger in mass due to transference of materialfrom the vapor onto the solid surface The deposition is normally carriedout in a vacuum chamber to enable control of the vapor composition If thevapor is created by physical means without a chemical reaction, the process

im-is classi¯ed as PVD; if the material deposited im-is the product of a chemicalreaction, the process is classi¯ed as CVD Many variations of these basicvapor deposition methods have been developed in e®orts to balance advan-tages and disadvantages of various strategies based on the requirements of

¯lm purity, structural quality, the rate of growth, temperature constraints

Fig 1.2 Schematic showing the basic features of evaporative deposition system.

and other factors In this section, the salient features of these processingmethods are brie°y described This is of general interest because the state

of stress in a ¯lm can be strongly in°uenced by its deposition history, asdescribed in the later sections of this chapter

1.2.1 Physical vapor depositionPhysical vapor deposition is a technique whereby physical processes, such

as evaporation, sublimation or ionic impingement on a target, facilitate thetransfer of atoms from a solid or molten source onto a substrate Evaporationand sputtering are the two most widely used PVD methods for depositing

¯lms

Figure 1.2 schematically illustrates the basic features of evaporativedeposition In this process, thermal energy is supplied to a source from whichatoms are evaporated for deposition onto a substrate The vapor source con-

¯guration is intended to concentrate heat near the source material and toavoid heeding the surroundings Heating of the source material can be ac-complished by any of several methods The simplest is resistance heating of

a wire or stripe of refractory metal to which the material to be evaporated

is attached Larger volumes of source material can be heated in crucibles

of refractory metals, oxides or carbon by resistance heating, high frequencyinduction heating, or electron beam evaporation The evaporated atomstravel through reduced background pressure p in the evaporation chamberand condense on the growth surface The deposition rate _R of the ¯lm iscommonly denoted by the number of atoms arriving at the substrate per

unit area of the substrate per unit time, by the time required to deposit

a full atomic layer of ¯lm material, or by the average normal speed of thegrowth surface of the ¯lm The deposition rate or °ux is a function of thetravel distance from the source to the substrate, the angle of impingementonto the substrate surface, the substrate temperature Ts, and the base pres-sure p If the source material (such as Cr, Fe, Mo, Si and Ti) undergoessublimation, su±ciently large vapor pressures may be obtained below itsmelting temperature so that a solid source could be employed for evapora-tive deposition On the other hand, for most metals in which a su±cientlylarge vapor pressure (» 10−3 torr, or 0.13 Pa) cannot be achieved at orbelow the melting temperature, the source is heated to a liquid state so as

to achieve proper deposition conditions

Metal alloys, such as Al{Cu, Co{Cr or Ni{Cr, can generally be orated directly from a single heated source If two constituents of the alloyevaporate at di®erent rates causing the composition to change in the melt,two di®erent sources held at di®erent temperatures may be employed toensure uniform deposition Unlike metals and alloys, inorganic compoundsevaporate in such a way that the vapor composition is usually di®erent fromthat of the source The resulting molecular structure causes the ¯lm sto-ichiometry to be di®erent from that of the source High purity ¯lms ofvirtually all materials can be deposited in vacuum by means of electronbeam evaporation

evap-Molecular beam epitaxy (MBE) is an example of an evaporative method.This growth technique can provide ¯lm materials of extraordinarily goodquality which are ideal for research purposes However, the rate of growth

is very low compared to other methods, which makes it of limited use forproduction of devices In MBE, the deposition of a thin ¯lm can be accu-rately controlled at the atomic level in an ultra-high vacuum (10−10torr, or1.33£10−8 Pa) A substrate wafer is placed in the ultra-high vacuum cham-ber It is sputtered brie°y with a low energy ion beam to remove surfacecontamination This step is followed by a high temperature anneal to relaxany damage done to the growth surface during preparation The substrate

is then cooled to the growth temperature, typically between 400 and 700◦C,and growth commences by directing atomic beams of the ¯lm material, aswell as a beam of dopant material if necessary, toward the growth surface ofthe substrate The beams are emitted from crucibles of the growth materialswhich have been heated to temperatures well above the substrate tempera-ture to induce evaporation and condensation The ¯lms may be examined

by transmission electron microscopy or x-ray di®raction after cooling Thecomplete history of evolution of internal stress in the ¯lm during deposi-

tion can be obtained in situ by monitoring the changes in curvature of thesubstrate on which the ¯lm is deposited as described in detail in Chapter 2.

Sputter gas

target(cathode)

substrate(anode)

glow discharge (Ar+) DC

voltagesource (or)

Fig 1.3 Schematic showing the basic features of a dc sputter deposition system.

In sputter deposition, ions of a sputtering gas, typically Ar, are erated toward the target at high speed by an imposed electric ¯eld The ini-tial concentration of charge carriers in the system is signi¯cantly increasedwith an increase in the dc voltage, as the ions collide with the cathode,thereby releasing secondary electrons, and with the neutral gas atoms Ascritical numbers of electrons and ions are created through such avalanches,the gas begins to glow and the discharge becomes self-sustaining Gaseousions striking the target or the source material from which the ¯lm is madedislodge surface atoms which form the vapor in the chamber The target isreferred to as the cathode since it is connected to the negative side of thedirect current power supply Figure 1.3 schematically shows the basic ele-ments of a sputter deposition system The chamber is evacuated and then

accel-Ar gas, at a pressure of approximately 13.3 Pa (10−1 torr), is introduced forthe purpose of maintaining a visible glow discharge The Ar+ions bombardthe target or cathode, and the ensuing momentum transfer causes the neu-

tral atoms of the target source to be dislodged These atoms transit throughthe discharge and condense onto the substrate, thus providing ¯lm growth.Several di®erent sputtering methods are widely used for the deposi-tion of thin ¯lms in di®erent practical applications: (i) dc sputtering (alsocommonly referred to as cathodic or diode sputtering), (ii) radio frequency(rf) sputtering with frequencies typically in the 5{30 MHz range, (iii) mag-netron sputtering, where a magnetic ¯eld is applied in superposition with aparallel or perpendicularly oriented electric ¯eld between the substrate andthe target source, and (iv) bias sputtering, where either a negative dc or rfbias voltage is applied to the substrate so as to vary the energy and °ux ofthe incident charged species.

There are many distinctions between the sputtering process and theevaporative process for ¯lm deposition, as described by Ohring (1992) forexample Evaporation is a thermal process where the atoms of the material

to be deposited arrive at the growth surface with a low kinetic energy Insputtering, on the other hand, the bombardment of the target source by Ar+ions imparts a high kinetic energy to the expelled source atoms Althoughsputter deposition promotes high surface di®usivity of arriving atoms, italso leads to greater defect nucleation and damage at the deposition surfacebecause of the high energy of the atoms While evaporation occurs in ahigh vacuum (10−6 to 10−10 torr, or 1.33£10−4 to 1.33£10−8 Pa), sput-tered atoms transit through a high pressure discharge zone with a pressure

of approximately 0.1 torr (13.33 Pa) Sputter-deposited ¯lms generally tain a higher concentration of impurity atoms than do ¯lms deposited byevaporation, and are prone to contamination by the sputtering gas As aresult, sputter deposition is not well suited for epitaxial growth of ¯lms.For polycrystalline ¯lms, the ¯lm grain structure resulting from sput-ter deposition typically has many crystallographic orientations without pre-ferred texture However, evaporative deposition leads to highly textured

con-¯lms for which the grain size is typically greater than that of the sputtered

¯lms Sputter deposition o®ers better control in maintaining stoichiometryand ¯lm thickness uniformity than evaporative deposition, and has the °exi-bility to deposit essentially any crystalline and amorphous materials Theseissues are discussed in more detail in Section 1.8

1.2.2 Chemical vapor depositionChemical vapor deposition is a versatile deposition technique that provides

a means of growing thin ¯lms of elemental and compound semiconductors,metal alloys and amorphous or crystalline compounds of di®erent stoichiom-

film deposit reactant gas

be-Si substrates, where pyrolysis at 650◦C leads to the decomposition of silanegas according to the reaction

SiH4(g) ! Si(s) + 2H2(g)

High-temperature reduction reactions where hydrogen gas is used as a ducing agent are also employed to produce epitaxial growth of Si ¯lms onmonocrystalline Si substrates at 1200◦C according to the reaction

re-SiCl4(g) + 2H2(g)! Si(s) + 4HCl(g)

The nature of epitaxy is described in detail later in this chapter

In CVD, as in PVD, vapor supersaturation a®ects the nucleation rate

of the ¯lm whereas substrate temperature in°uences the rate of ¯lm growth.These two factors together in°uence the extent of epitaxy, grain size, grainshape and texture Low gas supersaturation and high substrate tempera-tures promote the growth of single crystal ¯lms on substrates High gassupersaturation and low substrate temperatures result in the growth of lesscoherent, and possibly amorphous, ¯lms Low-pressure CVD (LPCVD),plasma-enhanced CVD (PECVD), laser-enhanced CVD (LECVD) and met-alorganic CVD (MOCVD) are variants of the CVD process used in manysituations to achieve particular objectives

1.2.3 Thermal spray deposition

powder injection internal external water

or solid state

substrate

air or vacuum chamber

Fig 1.5 Schematic illustration of the thermal spray process.

The thermal spray process of thin ¯lm fabrication refers broadly to arange of deposition conditions wherein a stream of molten particles impingesonto a growth surface In this process, which is illustrated schematically inFigure 1.5, a thermal plasma arc or a combustion °ame is used to melt andaccelerate particles of metals, ceramics, polymers or their composites to highvelocities in a directed stream toward the substrate The sudden deceleration

of a particle upon impact at the growth surface leads to lateral spreading andrapid solidi¯cation of the particle forming a `splat' in a very short time Thecharacteristics of the splat are determined by the size, chemistry, velocity,degree of melting and angle of impact of the impinging droplets, and by thetemperature, composition and roughness of the substrate surface Successiveimpingement of the droplets leads to the formation of a lamellar structure

in the deposit The oxidation of particles during thermal spray of metalsalso results in pores and contaminants along the splat boundaries Quenchstresses and thermal mismatch stresses in the deposit are partially relieved

by the formation of microcracks or pores along the inter-splat boundariesand by plastic yielding or creep of the deposited material There are severaldi®erent types of thermal spray processes, a review of which can be found

in Herman et al (2000)

Continuous or step-wise gradients in composition through the ness of the layer can be achieved by use of multiple nozzles whereby the

thick-Fig 1.6 Scanning electron micrographs showing the microstructures of spray coating of NiCrAlY on a 1020 steel substrate (a) Air plasma spray coated layer with inter-splat cracks whose origin can be traced to the oxidation of Al in the coated material during deposition (b) Vacuum plasma sprayed coating of the same material without inter-splat microcracks Reproduced with permission from Alcala et al (2001).)

plasma-°ow rate of the constituent phases of the deposited composite can be ulated during spray, as demonstrated by Kesler et al (1997) Alternatively,the feed rate of the powders of the di®erent constituent phases can also

mod-be controlled appropriately during thermal spray so as to deposit a gradedlayer onto a substrate Gradients in porosity can be introduced through thethickness of the deposited layer by manipulating the processing parametersand deposition conditions

The plasma spray technique o®ers a straightforward and cost-e®ectivemeans to spray deposits of metals and ceramics that are tens to hundreds ofmicrometers in thickness onto a variety of substrates in applications involv-ing thermal-barrier or insulator coatings Typical plasma-spray deposits areporous, with only 85{90% of theoretical density

For applications requiring higher density coatings with a strong sion to the substrate, low-pressure plasma spray is employed where spraying

adhe-is done in an inert-gas container operating at a reduced pressure Vacuumplasma spray is another thermal spray process which is used to improvepurity of the deposited material and to reduce porosity and defect content,albeit at a higher cost than air plasma spray

Figure 1.6(a) is a representative micrograph of the cross-section of anair-plasma-sprayed NiCrAlY coating, commonly used as a bondcoat between

a ceramic thermal barrier coating and a nickel-base superalloy substrate ingas turbine engines, as described in the example in the next subsection Thiscoating was deposited onto a 1020 steel substrate The dark streaks arethe inter-splat boundaries along which microcracks and voids have formed.The origins of these defects could be traced to the formation of Al2O3 dur-ing deposition (Alcala et al 2001) On the other hand, vacuum plasmaspray deposition of the same material onto the steel substrate results in thesuppression of such oxidation and the attendant cracking of the inter-splatboundaries, as shown in Figure 1.6(b) The resulting coating has a moreuniform microstructure with a signi¯cantly reduced pore density

1.2.4 Example: Thermal barrier coatings

The thermal barrier coating (TBC) system is a multilayer arrangement introduced

to thermally insulate metallic structural components from the combustion gases in gas turbine engines The design of TBCs along with appropriate internal cooling

of the high-temperature metallic components has facilitated the operation of gas turbine engines at gas temperatures well in excess of the melting temperature of the turbine blade alloy This thermal protection system, which reduces the surface temperature of the alloy by as much as 300 ± C, leads to better engine e±ciency, performance, durability and environmental characteristics.

The performance requirements for TBCs are stringent The selection of materials for TBCs and the design of the layered coating structure inevitably re- quires consideration of highly complicated interactions among such phenomena and processes as phase transformation, microstructural stability, thermal conduction, di®usion, oxidation, thermal expansion mismatch between adjoining materials, ra- diation, as well as damage and failure arising from interface delamination, ¯lm buckling, subcritical fracture, foreign-object impact, erosion, thermal and mechan- ical fatigue, inelastic deformation and creep Summaries and reviews of such issues for TBCs have been reported by Evans et al (2001) and Padture et al (2002).

A representative TBC system for a gas turbine engine comprises four layers: (a) a metallic substrate, that is, the turbine blade itself, (b) a metallic interlayer or bondcoat, (c) a thermally grown oxide (TGO), and (d) a ceramic outerlayer or top- coat A schematic of the turbine blade along with a scanning electron micrograph

of the cross-section of a layered TBC coating is shown in Figure 1.7.

The turbine blade is commonly made of a nickel-base or cobalt-base

super-Fig 1.7 Illustration of the thermal-barrier coated turbine blade which is air-cooled internally along hollow channels The outer surface of the blade is coated for thermal protection from the hot gases so that there exists a temperature gradient through the cross-section of the TBC Also shown is a scanning electron micrograph

of the cross-sectional view of the coating layers which comprise a ceramic topcoat deposited by electron-beam PVD, an alumina TGO layer, and a NiCrAlY bondcoat

on a nickel-base superalloy substrate Reprinted with permission from Padture et

al (2002).)

alloy which is investment-cast as a single crystal or polycrystal The bondcoat, typically 75 to 150 ¹m in thickness and made of an oxidation-resistant alloy of NiCrAlY or NiCoCrAlY, is deposited onto the substrate using plasma spray or electron-beam PVD In some cases, the bondcoats are deposited by electroplating along with either di®usion-aluminizing or CVD with layers of Ni and Pt aluminides Occasionally, the bondcoats consist of sublayers of di®erent phases or compositions The bondcoat is a critical component of the TBC system that determines the spal- lation resistance of the TBC.

The oxidation of the bondcoat at an operating temperature as high as 700 ± C results in a 1 to 10 ¹m thick TGO layer between the bondcoat and the topcoat This oxidation process is aided by the transport of oxygen from the surrounding hot gases in the engine environment through the porous TBC top layer The TGO layer

is also commonly engineered to serve as a di®usion barrier so as to suppress further oxidation of the bondcoat; for this purpose, the in-situ formation of a uniform, defect-free ®-Al 2 O 3 TGO interlayer is facilitated by controlling the composition of the bondcoat.

Y 2 O 3 -stabilized Zr 2 O 3 (YSZ), typically with a Y 2 O 3 concentration of 7 to 8 wt%, is the common material of choice for the ceramic topcoat in light of its many desirable properties as a TBC (Padture et al 2002):

¡ the thermal expansion coe±cient of YSZ has a high value of approximately

11 £10 ¡6 = ± C, which is closer to that of the metallic layer beneath it ( »

14 £ 10 ¡6 = ± C) than to that of most ceramic materials Consequently, the stresses generated by thermal expansion mismatch with the underlying layers during the thermal cycles generated by the operation of the turbine engine would be minimized;

¡ YSZ has a very low thermal conductivity of approximately 2.3 W/m¢K at

1000±C when fully dense;

¡ the low density of YSZ, typically about 6:4 £ 10 3 kg/m 3 , improves the formance of the rotating engine component in which it is used as a coating;

per-¡ the high melting temperature of approximately 2700 ± C makes YSZ a able thermal barrier material;

desir-¡ with a myriad of available processing methods, the YSZ topcoat can be deposited with controlled pore content and distribution in a such a way that

it can be made compliant with an elastic modulus of approximately 50 GPa; and

¡ YSZ has a hardness of roughly 14 GPa which renders it resistant to damage from foreign object impact and erosion.

The two most common methods of producing the TBC topcoat entail air plasma spray (see Section 1.2.3) and electron-beam physical vapor deposition, EB- PVD (see Section 1.2.1) The former deposition method produces 15 to 25% poros- ity whereby low values of thermal conductivity and elastic modulus result The spray coating, typically 300 to 600 ¹m thick, contains pancake-shaped splats with

a diameter of 200{400 ¹m and thickness of 1{5 ¹m The pores and cracks along the inter-splat boundaries are generally oriented parallel to the interface with the bondcoat and normal to the direction of heat °ow, a consequence of which is the low thermal conductivity of 0.8{1.7 W/m ¢K The air plasma spray method provides an economical means for the large-scale production of TBCs However, the structural defects inherent in this process and the rough interface between the sprayed coat- ing and the material beneath it make this deposition technique more suitable for less critical parts The EB-PVD deposited TBCs, on the other hand, are typically

125 ¹m thick and are more durable and costly compared to the plasma-sprayed coatings They are engineered to comprise the following microstructural features: (a) an equiaxed structure of YSZ grains with a diameter of 0.5{1.0 ¹m near the in- terface with the bondcoat, (b) columnar grains of YSZ, 2{10 ¹m in diameter which extend from the equiaxed grains to the top surface of the coating, with the bound- aries between the columnar grains amenable to easy separation to accommodate thermal stresses that develop during service, and (c) nanometer-size pores inside the columnar grains.

1.3 Modes of film growth by vapor deposition

There is enormous variation in the microstructures of ¯lms formed by position of atoms on the surfaces of substrates from vapors Final struc-tures can range from single crystal ¯lms, through polycrystalline ¯lms withcolumnar or equiaxed grains, to largely amorphous ¯lms Some materialscan be deposited in ways that yield any of these structures, with the ¯nalmicrostructure depending on the materials involved, the deposition methodused and the environmental constraints imposed The purpose in this sec-tion is to discuss some general ideas in ¯lm growth by vapor deposition thattranscend issues of ¯nal microstructure The discussion is largely descrip-tive, and it is couched in the terminology of thermodynamics The principles

de-of thermodynamics provide powerful tools for deciding whether or not someparticular change in a material system can occur On the other hand, inthose cases in which the change considered can indeed take place, thermo-dynamics is silent on whether or not it does take place and, if it does occur,

on how it proceeds Nonetheless, the thermodynamic framework provides

a basis for establishing connections between deposition circumstances and

¯lm formation Progress toward understanding physical processes of ¯lmgrowth are discussed in depth by Tsao (1993), Pimpinelli and Villain (1998)and Venables (2000)

1.3.1 From vapor to adatoms

In this section, the factors that control the very early stages of growth of athin ¯lm on a substrate are described in atomistic terms The process beginswith a clean surface of the substrate material, which is at temperature Ts,exposed to a vapor of a chemically compatible ¯lm material, which is at thetemperature Tv To form a single crystal ¯lm, atoms of the ¯lm material inthe vapor must arrive at the substrate surface, adhere to it, and settle intopossible equilibrium positions before structural defects are left behind thegrowth front To form an amorphous ¯lm, on the other hand, atoms must

be prevented from seeking stable equilibrium positions once they arrive atthe growth surface In either case, this must happen in more or less thesame way over a very large area of the substrate surface for the structure

to develop At ¯rst sight, this outcome might seem unlikely, but such ¯lmsare produced routinely

Atoms in the vapor come into contact with the substrate surface where

Fig 1.8 Schematic showing the atomistics of ¯lm formation on substrates.

they form chemical bonds with atoms in the substrate The temperature ofthe substrate must be low enough so that the vapor phase is supersaturated

in some sense with respect to the substrate, an idea that will be made moreconcrete below There is a reduction in energy due to formation of the bondsduring attachment Some fraction of the attached atoms, which are calledadatoms, may return to the vapor by evaporation if their energies due tothermal °uctuations are su±cient to occasionally overcome the energy ofattachment, as suggested in the schematic diagram in Figure 1.8 To makethe discussion a bit more speci¯c, a simple hexagonal close-packed crystalstructure is assumed for convenience in counting bonds

It has already been recognized that, for ¯lm growth to be possible, it isnecessary that the vapor in contact with the growth surface is supersaturatedwith respect to the substrate at its temperature Ts For a homogeneouscrystal at some temperature that is in contact with its own vapor at thesame temperature, the equilibrium vapor pressure peof the system is de¯ned

as the pressure at which condensation of vapor atoms onto the solid surfaceand evaporation of atoms from the surface proceed at the same rate Atequilibrium, the entropic free energy per atom in the vapor equals the freeenergy per atom in the interior of the crystal The lower internal energy ofatoms in the interior of the crystal compared to those in the vapor, due tochemical bonding, is o®set by the lower entropic energy within the crystal.For net deposition on the substrate surface, it is essential that the pressure

p in the vapor exceeds the equilibrium vapor pressure pe at the substratetemperature, that is, the vapor must be supersaturated For the pressure

p, the entropic free energy per atom of the vapor, over and above the freeenergy at pressure pe, is estimated as the work needed per atom to increasethe vapor pressure from pe to p at constant temperature According to theideal gas law, the result is kTvln(p=pe) where k = 1:38£10−23J/K= 8:617£

10−5eV/K is the Boltzmann constant and Tv is the absolute temperature

of the vapor If the vapor becomes supersaturated, a free energy di®erencebetween the vapor and the interior of the crystal exists, providing a chemicalpotential for driving the advance of the interface toward the vapor As theinterface advances in a self-similar way, a remote layer of vapor of some mass

is converted into an interior layer of crystal of the same mass The interfacedoes not advance or retreat when these energies are identical

In ¯lm deposition, the situation is often complicated by the fact thatthe vapor and the substrate are not phases of the same material, and bythe fact that the temperature of the substrate is usually lower than that ofthe vapor The de¯nition of equilibrium vapor pressure is not so clear inthis case However, in most cases there will be some level of vapor pressurebelow which deposition of ¯lm material onto the growth surface will notoccur; this serves as an operational de¯nition of pe for ¯lm growth

Once adatoms become attached to the substrate, their entropic freeenergy is reduced from that of the vapor The adatoms form a distribution

on the substrate surface having the character of a two-dimensional vapor.The deposited material it is believed to thermalize quickly and to take on thetemperature Ts On a crystal surface, there is some density ½ad of adatomsthat is in equilibrium with a straight surface step or ledge bordering a partialmonolayer of ¯lm atoms, that is, for which the step neither advances norrecedes The step is a boundary between two phases in a homogeneousmaterial system Consequently, the notion of equilibrium vapor pressure

or equilibrium vapor density ½e

ad can be invoked in an analogous way For

¯lm growth to occur by condensation, the free energy per atom of the dimensional gas must exceed the free energy of a fully entrained surfaceatom by the amountEph= kTsln(½ad=½e

two-ad)

1.3.2 From adatoms to film growthEach adatom presumably resides within an equilibrium energy well on thesurface most of the time, and this well is separated from adjacent energywells by a barrier of height Ed > 0 with respect to the equilibrium po-sition The atoms oscillate in their wells due to thermal activation and,occasionally, they acquire su±cient energy to hop into adjacent equilibriumwells Surface di®usion results from such jumps If the surface is spatiallyuniform, di®usion occurs randomly with no net mass transport on a macro-scopic scale The hopping rate of any given atom increases with increasingsubstrate temperature Occasionally, attached atoms may also change po-sitions with substrate atoms, but this possibility is not pursued further in

this discussion If the surface is not uniform, perhaps as a result of a straingradient or a structural gradient, then surface di®usion can be directionallybiased, resulting in a net mass transport along the surface on a macroscopicscale Consequences of such transport are considered in Chapters 8 and 9 Ifthe temperature is very low or if the di®usion barrier is very high, adatomsstick on the growth surface where they arrive, and the ¯lm tends to growwith an amorphous or very ¯ne-grained polycrystalline structure.

The growth surface invariably has some distribution of surface defects{ crystallographic steps, grain boundary traces and dislocation line termi-nations, for example { which provide sites of relatively easy attachment foradatoms The spacing between defects represents a length scale for compar-ison to the extent of random walk di®usion paths If the di®usion distance islarge compared to the defect spacing, then adatoms tend to encounter thesedefects and become attached to them, giving up some free energy in theprocess This is the case of heterogeneous nucleation and growth of ¯lms.General statements about such processes are also di±cult to make In somecases, surface steps are relatively transparent to migrating adatoms, that is,adatoms often pass over the step in either the up direction or the down direc-tion without attachment For other materials, virtually every encounter of

an adatom with the step results in attachment In yet other cases, adatomseasily bypass steps in the up direction but not in the down direction, a mani-festation of the so-called Schwoebel barrier (Schwoebel 1969) If the spacing

of defects is large compared to the di®usion distance, on the other hand,then migrating adatoms have the potential for lowering the energy of thesystem by binding together upon mutual encounters to form clusters Thecase of formation of such stable clusters is called homogeneous nucleation

of ¯lm growth That there is some minimum cluster size necessary for mation of a stable nucleus can be demonstrated by appeal to the followingargument based on classical nucleation theory

for-It was noted in Section 1.3.1 that the free energy of an adatom in thesupersaturated distribution on the surface is Eph with respect to its stateonce entrained within the surface layer In other words, this is the amount

of energy reduction per atom associated with the phase change from a twodimensional gas of adatoms on the surface to a completely condensed surfacelayer Suppose that a planar cluster of n atoms is formed on the surface.Will it tend to grow into a larger cluster, and eventually into a ¯lm, or willthe cluster tend to disperse? If all n atoms are fully entrained within thecluster then the free energy reduction due to cluster formation would be

¡nEph However, those atoms on the periphery of the cluster are not fullyincorporated They possess an excess free energy compared to those that

are fully entrained; this energy can be estimated to be roughly 4p

¼nEf for

an equiaxed cluster, if n is fairly large compared to unity This quantity hasthe character of a surface step energy The free energy change associatedwith cluster formation is then

¢F = ¡nEph+ 4p

A graph of F versus n for ¯xed ¢Eph and Ef shows a maximum value for

a cluster size of n∗ = 4¼(Ef=Eph)2 The value of ¢F at n∗, which is theactivation energy for cluster formation, is ¢F∗ = 4¼E2

f=Eph The implication

is that clusters smaller than the size n∗ are unstable and that they tend todisperse, whereas clusters larger than this size tend to grow, driven by acorresponding reduction in free energy It is likely that many clusters formand disperse for each one that evolves into an island In fact, the classicalnucleation theory says nothing about such processes, other than that theyare possible In any case, for a ¯lm to form on the substrate surface, it isnecessary that either nuclei formed by such homogeneous processes are able

to grow or that a su±cient number of surface defects are available to serve

as sites of heterogeneous nucleation

The mode of ¯lm formation is determined by the relative values of thevarious energies involved in the process, and this mode largely determinesthe eventual structure of the ¯lm There are two main comparisons to beconsidered One of these contrasts the height of the di®usion barrier Ed tothe background thermal energy IfEd is large compared to the backgroundthermal energy then surface mobility of adatoms is very low Under suchconditions, adatoms more or less stick where they arrive on the substratesurface

For growth of crystalline ¯lms, it is important that Ed be less thanthe background thermal energy so that adatoms are able to seek out andoccupy virtually all available equilibrium sites in the ¯lm crystal lattice

as it grows This requires the substrate temperature and/or the degree ofsupersaturation of the vapor to be high enough to insure such mobility.Suppose that this is indeed so and that the adatoms are able to migrateover the surface

The other important energy comparison concerns the propensity foratoms of ¯lm material to bond to the substrate This is represented by themagnitude ofEfs, relative to their tendency to bond to other, less well-bound,atoms of ¯lm material, as represented byEf Two kinds of growth processescan be distinguished, one withEfslarger in magnitude thanEf, and a secondwith the relative magnitudes reversed If Efs is the larger of the two energychanges in magnitude, then ¯lm growth tends to proceed in a layer by layer

mode, as indicated in the schematic diagram in Figure 1.8 Adatoms aremore likely to attach to the substrate surface than to other ¯lm materialsurfaces Once small stable clusters of adatoms form on the surface, otheradatoms tend to attach to the cluster at its periphery where they can bondwith both substrate and ¯lm atoms, thereby continuing the planar growth,

as indicated in Figure 1.8 This layer-by-layer ¯lm growth mode is oftencalled the Frank—van der Merwe growth mode or FM mode, according to acategorization of growth modes proposed by Bauer (1958) on the basis ofmore macroscopic considerations of surface energy This alternate point ofview will be considered in Section 1.3.5

On the other hand, if Ef is larger in magnitude than Efs, then it isenergetically favorable for adatoms to form three-dimensional clusters orislands on the surface of the substrate Film growth proceeds by the growth

of islands until they coalescence; this type of growth is commonly called theVolmer—Weber growth modeor the VW mode; see Figure 1.8

A third type of growth, which combines features of both the Frank{vander Merwe and the Volmer{Weber modes, is called the Stranski—Krastanovgrowth mode or SK mode In this mode, the ¯lm material tends to preferattachment to the growth surface rather than the formation of clusters onthe growth surface; that is, Efs is greater in magnitude than Ef However,after a few monolayers of ¯lm material are formed and after the structure ofthe ¯lm becomes better de¯ned as a crystal in conformity with the substrate,the tendency is reversed In other words, once the planar growth surfacebecomes established as ¯lm material, subsequent adatoms tend more togather into clusters than to continue planar growth The magnitude ofEfsappears to depend on the thickness of the ¯lm in the early stages of growth,decreasing from values larger than the magnitude of Ef to values that aresmaller The occurrence of this mode is most likely when the ¯rst few layers

of ¯lm material are heavily strained due to the constraint of the substrate.This issue is revisited in Section 1.3.5

1.3.3 Energy density of a free surface or an interface

In the preceding discussion, the early stages in the growth of a ¯lm from avapor were considered in a qualitative way in terms of the behavior of atoms.Many of the inferences drawn can also be made in terms of surface energyand interface energy of solid materials These quantities are macroscopicmeasures attributable to the discreteness of the material However, theyrepresent only ensemble averages of behavior at the atomistic level and donot incorporate discreteness of the material in any direct way In many